PreCalculus - Matrices & Matrix Applications (8 of 33) Gaussian Elimination: 3x3 Matrix, No Solution

Michel van Biezen
10 Jun 201505:43
EducationalLearning
32 Likes 10 Comments

TLDRThis video tutorial demonstrates the Gaussian elimination method for solving a 3x3 system of equations, highlighting the process when there is no solution. It explains that the absence of a solution corresponds to three planes in space that do not intersect, as exemplified by the final row of the augmented matrix indicating an impossible equation (0x + 0y + 0z = -3). The explanation is clear and methodical, showing each step of the elimination process and its implications for the solvability of the system.

Takeaways
  • 📌 The Gaussian elimination method is used to solve a system of linear equations with three unknowns (x, y, z).
  • 📍 In this video, the focus is on demonstrating how to handle a situation where there is no solution for the given system of equations.
  • 🔍 The script explains that three equations represent three planes in the XYZ Cartesian coordinate system, and a solution exists if there's a single point where all three planes intersect.
  • ⏹ When two of the planes are parallel, they will never intersect, leading to a situation with no solution.
  • 🧩 The process begins by writing the augmented matrix, combining the coefficients of x, y, & z on one side and the constants on the other.
  • 🔄 The goal of the Gaussian elimination is to get a diagonal of 1s to easily read the values for x, y, & z.
  • 📈 The script details the step-by-step process of eliminating variables from rows below the pivot to achieve the diagonal of 1s.
  • 🤔 The realization that there's no solution occurs when the final matrix results in an impossible equation like '0x + 0y + 0z = -3'.
  • ❌ An impossible equation, such as 0 equals a negative number, indicates that there's no set of values for x, y, & z that can satisfy all three equations simultaneously.
  • 📊 The video serves as an educational tool to understand not only how to apply Gaussian elimination but also how to recognize when a system of equations has no solution.
Q & A

  • What is the Gaussian method of elimination used for in this context?

    -In this context, the Gaussian method of elimination is used to solve a system of three equations with three unknowns, specifically to find the x, y, and z values that satisfy all three equations simultaneously.

  • How does the video script relate the equations to geometric planes in a Cartesian coordinate system?

    -The video script explains that each of the three equations represents a plane in the XYZ Cartesian coordinate system. Finding a solution corresponds to finding a point in space where all three planes intersect, which is represented by an XYZ coordinate value.

  • What is the significance of the three planes being angled in a certain way?

    -The significance of the three planes being angled in a certain way is that they must intersect at a single point in space for there to be a solution to the system of equations. If the planes are parallel or do not intersect at a single point, there will be no solution.

  • What is an example of a situation where there is no solution?

    -An example of a situation where there is no solution is when two of the planes are parallel to each other. Parallel planes will never intersect, and therefore, there is no common point in space that satisfies all three equations simultaneously.

  • How does the Augmented matrix represent the system of equations?

    -The Augmented matrix represents the system of equations by placing the coefficients of x, y, and z on the left side in the form of a matrix, with each row corresponding to one of the equations. The constants from the equations are placed on the right side of the matrix, separated by a line.

  • What is the goal of the Gaussian elimination method in this context?

    -The goal of the Gaussian elimination method in this context is to manipulate the Augmented matrix to a form where the diagonal elements are all 1's and the elements below the diagonal are 0's. This form allows for the direct reading of the x, y, and z values that solve the system of equations.

  • What operation is performed to eliminate the 2 in the third row of the matrix?

    -To eliminate the 2 in the third row, the second row (which has a 1 in the pivot position) is multiplied by -2 and added to the third row. This operation results in a 0 in the third position of the second row and a -2 in the third position of the third row.

  • Why is it necessary to have a 1 in the pivot position?

    -Having a 1 in the pivot position simplifies the elimination process. It allows for the direct subtraction or addition of other rows to eliminate variables from the pivot column without the need for additional multiplication or division operations.

  • What does it mean when the final matrix results in a situation where 0x + 0y + 0z = -3?

    -When the final matrix results in a situation where 0x + 0y + 0z = -3, it indicates that there is no solution to the system of equations. This is because it is impossible for the sum of three zero values to equal -3, which contradicts the fundamental principles of arithmetic.

  • How does the video script conclude the demonstration?

    -The video script concludes by demonstrating that when the final matrix results in an impossible situation (0 = -3), it confirms that there is no solution to the system of equations. This means that there is no common point in space where all three planes intersect.

  • What is the primary takeaway from the video script?

    -The primary takeaway from the video script is understanding how the Gaussian elimination method can be used to determine whether a system of three equations with three unknowns has a solution, and how the method reveals when there is no solution due to the impossibility of the given equations.

Outlines
00:00
📚 Introduction to Gaussian Elimination and No Solution Scenario

This paragraph introduces the concept of using the Gaussian elimination method to solve a system of three equations with three unknowns (x, y, and z) represented by a 3x3 matrix. It explains that the solution involves finding a common point in space where three planes intersect, which corresponds to the values of x, y, and z that satisfy all equations simultaneously. The paragraph also sets up the context for discussing a scenario where no solution exists, such as when two of the planes are parallel and do not intersect, hence no common point can be found. The explanation begins with the construction of an augmented matrix, detailing the placement of coefficients and constants, and the process of elimination to reach a solution or identify the absence thereof.

05:01
🔍 Analyzing the No Solution Outcome with Gaussian Elimination

This paragraph delves into the outcome of using Gaussian elimination when there is no solution to the system of equations. It describes the process of transforming the augmented matrix to isolate the variables x, y, and z, and how the elimination leads to the realization that no solution exists. The paragraph highlights the conclusion that zero cannot equal a negative number (negative 3 in this case), which indicates an inconsistency in the equations. This inconsistency means that there is no set of values for x, y, and z that can satisfy all three equations at once, confirming that a common point in space for the three planes does not exist, and thus, no solution is possible.

Mindmap
Keywords
💡Gaussian Elimination
Gaussian Elimination is a systematic method used to solve systems of linear equations by transforming the coefficient matrix into an upper triangular form through a series of row operations. In the video, it is used to demonstrate the process of solving a 3x3 matrix, which represents three equations with three unknowns (x, y, and z). The method helps to identify whether a unique solution exists or if the system has no solution.
💡3x3 Matrix
A 3x3 Matrix is a rectangular array of numbers with three rows and three columns. In the context of the video, this matrix represents a system of three linear equations with three variables (x, y, and z). Each row of the matrix corresponds to one of the equations, with the coefficients of the variables and a constant term.
💡Augmented Matrix
An Augmented Matrix is a matrix that combines the coefficient matrix of a system of linear equations with its constant terms, allowing for the visualization of both the system's structure and the solution process. In the video, the Augmented Matrix is used to apply the Gaussian Elimination method, with the left portion containing the coefficients of x, y, and z, and the right side containing the constants.
💡Pivot
In the context of Gaussian Elimination, a Pivot is the element in a matrix that is currently being used to transform the row or the matrix into a more simplified form. Pivots are chosen to be non-zero elements that facilitate the elimination of variables in the rows below or above them. The video explains the process of making the pivot 1 and then using it to eliminate other elements in the same column.
💡No Solution
In the context of systems of linear equations, 'No Solution' refers to the situation where there is no set of values for the variables that can satisfy all the equations simultaneously. This can occur when the equations represent conditions that are inherently contradictory or when the planes represented by the equations in a three-dimensional space do not intersect at a common point.
💡Row Operations
Row Operations are the mathematical processes performed on rows of a matrix during Gaussian Elimination to simplify the system of equations. These operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting one row from another. The goal is to create an upper triangular matrix that makes it easier to determine the system's solution.
💡Coefficients
Coefficients are the numerical factors that multiply the variables in a linear equation. They determine the slope or direction of the plane in a three-dimensional Cartesian coordinate system. In the video, the coefficients for x, y, and z are represented in the matrix and are crucial for solving the system of equations.
💡Variables
Variables are symbols, often letters like x, y, and z, that represent unknown quantities in a mathematical equation. In the context of the video, variables are used to find the values that satisfy the system of linear equations. The process of solving the equations involves manipulating the coefficients and constants to isolate the variables and find their values.
💡Parallel Planes
Parallel Planes are planes in a three-dimensional space that do not intersect or meet each other, no matter how far they are extended. In the context of the video, if the equations represent the planes, parallel planes would mean that there is no common point where all the planes intersect, leading to a system of equations with no solution.
💡XYZ Cartesian Coordinate System
The XYZ Cartesian Coordinate System is a coordinate system that uses three perpendicular axes, labeled X, Y, and Z, to define positions in three-dimensional space. Each axis represents a dimension, and the intersection of the axes forms the origin. In the video, the equations are interpreted as planes in this coordinate system, and the goal is to find the point where all three planes intersect at the same (x, y, z) coordinates.
💡Upper Triangular Form
Upper Triangular Form is a specific arrangement of a matrix where all the non-zero elements are located on or above the diagonal, with zeros below the diagonal. This form is often achieved through Gaussian Elimination and simplifies the process of back-substitution to find the solution to a system of linear equations.
Highlights

Introduction to using Gaussian elimination for solving a 3x3 matrix representing three equations with three unknowns.

Explanation of the physical representation of equations as planes in the XYZ Cartesian coordinate system.

Description of the conditions necessary for a single solution where all three planes intersect at one point.

Introduction of a scenario where no solution exists due to parallel planes.

Detailed walkthrough of setting up the augmented matrix with coefficients and constants separated by a line.

Step-by-step demonstration of achieving zeros below the leading 1 in the first column using elementary row operations.

Continuation of Gaussian elimination to achieve zeros above the leading ones.

Adjustment of rows to simplify the matrix further, aiming for diagonal ones across the matrix.

Introduction of a problematic scenario where an equation reduces to 0 = -3, indicating no possible solution.

Explanation of the implications when the resulting matrix leads to an impossible equation.

Conclusion that no set of X, Y, and Z values exists that allows the three planes to intersect at a common point.

Insight into the limitations of certain system configurations in Gaussian elimination.

Explanation of how Gaussian elimination helps identify the feasibility of solutions for systems of equations.

Discussion on the role of Gaussian elimination in understanding the geometry of equations in three-dimensional space.

Final statement on the outcome of the Gaussian elimination method when applied to systems with no solution.

Transcripts

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Gaussian EliminationLinear AlgebraNo SolutionMath Tutorial3x3 MatrixEquation SolvingMath EducationAlgebra ConceptsXYZ CoordinatesMath Problem